Where Did Exponents Come From?
Imagine you need to multiply 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. That's ten twos! Writing all of that out is slow and easy to mess up. Mathematicians faced the same problem hundreds of years ago. They invented exponents (a shorthand for repeated multiplication) so they could write large calculations in a tiny, clear way. Instead of ten twos, you just write 2¹⁰. Let's see how this idea developed over time.
So here is the big question this lesson answers: How do you read, evaluate, and simplify expressions that contain exponents — especially when other operations like addition, subtraction, and multiplication are mixed in?
Core Principles of Exponents
Before we start calculating, you need to know a few key ideas. These are the building blocks for every exponent problem you will see on the SSAT.
Base and Exponent
Exponent Means Repeated Multiplication
Special Exponents: 1 and 0
Negative Bases Need Parentheses
Order of Operations (PEMDAS)
Seeing Exponents in Action
The diagram below shows how quickly numbers grow when you raise them to higher powers. Notice how 2 to the first power is just 2, but 2 to the sixth power is already 64. Exponents make numbers grow very fast!
Look at the jump from 2⁵ = 32 to 2⁶ = 64. Every time the exponent goes up by 1, the value doubles. That's because you are multiplying by one more 2 each time. This is what makes exponents so powerful — and why you need to evaluate them carefully.
The Rules You Need to Know
Here are the key formulas and rules for evaluating exponents. You do not need a calculator for the SSAT — so practice doing these by hand!
Common Powers You Should Memorize
Since you cannot use a calculator on the SSAT, knowing the most common powers by heart will save you time. The table and diagram below show the perfect squares, perfect cubes, and a few other handy powers.
| Expression | Expanded Form | Value |
|---|---|---|
| 2² | 2 × 2 | 4 |
| 3² | 3 × 3 | 9 |
| 4² | 4 × 4 | 16 |
| 5² | 5 × 5 | 25 |
| 6² | 6 × 6 | 36 |
| 10² | 10 × 10 | 100 |
| 2³ | 2 × 2 × 2 | 8 |
| 3³ | 3 × 3 × 3 | 27 |
| 5³ | 5 × 5 × 5 | 125 |
| 10³ | 10 × 10 × 10 | 1,000 |
Try to memorize the perfect squares (1² through 12²) and the small perfect cubes (2³, 3³, 4³, 5³). Knowing these from memory means you can solve SSAT problems faster and more confidently.
Worked Examples: Step by Step
Example 1: Evaluating 3⁴
Example 2: Order of Operations with Exponents
Example 3: Negative Base with Parentheses
Common Mistakes and How to Avoid Them
Exponent problems aren't usually hard — but they are full of traps! Here are the most common mistakes students make, and how to dodge them.
| Common Mistake | What Students Do Wrong | The Correct Approach |
|---|---|---|
| Multiplying instead of raising | Think 2⁴ = 2 × 4 = 8 | 2⁴ = 2 × 2 × 2 × 2 = 16 |
| Forgetting PEMDAS order | In 3 + 2², add first: 5² = 25 | Exponents first: 3 + 4 = 7 |
| Ignoring parentheses with negatives | Think −3² = 9 | −3² = −9. Only (−3)² = 9 |
| Getting 0 exponent wrong | Think 5⁰ = 0 | 5⁰ = 1 (any nonzero number to the zero power is 1) |
Connection to More Advanced Math
Right now you are working with small whole-number exponents, like 2³ or 5⁴. But as you move into higher math, exponents will show up in many new ways. Here is a preview of what's ahead.
| What You Know Now | What You'll Learn Later | Example |
|---|---|---|
| Positive whole-number exponents (2³ = 8) | Negative exponents (means "1 divided by") | 2⁻³ = 1/8 |
| Squaring a number (5² = 25) | Square roots (the reverse of squaring) | √25 = 5 |
| Evaluating one exponent at a time | Exponent rules for multiplying and dividing powers | 2³ × 2⁴ = 2⁷ |
| Small numbers like 3⁴ = 81 | Scientific notation for huge and tiny numbers | 3.5 × 10⁶ = 3,500,000 |
You don't need to worry about these advanced topics for the SSAT Middle Level. But knowing that exponents are a building block for future math should motivate you to master them now. Everything you learn here will make those future topics much easier!
Practice Problems
Try these five problems on your own. They get harder as you go. No calculator — just like the real SSAT! After you pick your answer, read the full explanation.
Lesson Summary
An exponent is a shorthand for repeated multiplication. In the expression aⁿ, the base (a) is the number being multiplied, and the exponent (n) tells you how many times. Remember that any nonzero number to the zero power equals 1, and any number to the first power equals itself.
When you see exponents mixed with other operations, always follow PEMDAS: evaluate what's inside parentheses first, then exponents, then multiplication and division left to right, and finally addition and subtraction left to right. Watch out for negative bases — (−3)² = 9, but −3² = −9. Parentheses make all the difference! Memorize your common perfect squares and cubes to save time on the SSAT.